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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

A family of semistable elliptic curves with large Tate-Shafarevitch groups


Author: Kenneth Kramer
Journal: Proc. Amer. Math. Soc. 89 (1983), 379-386
MSC: Primary 14K07; Secondary 11G05, 14G25
MathSciNet review: 715850
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Abstract: We present a family of elliptic curves defined over the rationals $ {\mathbf{Q}}$ such that each curve admits only good or multiplicative reduction and for every integer $ n$ there is a curve whose Tate-Shafarevitch group over $ {\mathbf{Q}}$ has more than $ n$ elements of order 2. Previously known examples of large Tate-Shafarevitch groups were constructed by forcing many places of additive reduction.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0715850-1
PII: S 0002-9939(1983)0715850-1
Article copyright: © Copyright 1983 American Mathematical Society